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Robust Mesh Segmentation Using Feature-Aware Region Fusion^{ †}

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## Abstract

**:**

## 1. Introduction

- An efficient over-segmentation method is introduced via adaptive space partition.
- By defining a new intra-region difference, inter-region difference, and fusion condition, a simple but powerful feature-aware region fusion algorithm is proposed that can robustly achieve mesh segmentation.

## 2. Related Work

_{0}optimization problem with respect to the Fiedler vector. Zhang et al. [25] segmented a mesh by blending regions into different patches and fitted each patch through a surface primitive. Lin et al. [26] performed segmentation based on a medial axis transform, which encodes both geometrical and structural information.

_{0}constraint to locate segmentation boundaries.

## 3. Feature-Aware Mesh Segmentation Algorithm

#### 3.1. Efficient Adaptive Space Partition

#### 3.2. New Feature-Aware Region Fusion

#### 3.2.1. Feature Description

_{2}norm. In other words, the difference is the Euclidean distance between feature vectors.

#### 3.2.2. Region Fusion

#### 3.2.3. Boundary Smoothing

## 4. Experiments and Discussions

#### 4.1. Results and Analysis

#### 4.2. Robustness Evaluations

#### 4.3. Qualitative and Quantitative Comparisons

#### 4.4. Further Discussions and Time Performance

## 5. Conclusions and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Katz, S.; Tal, A. Hierarchical mesh decomposition using fuzzy clustering and cuts. ACM Trans. Graph.
**2003**, 22, 954–961. [Google Scholar] [CrossRef] - Au, O.K.; Zheng, Y.; Chen, M.; Xu, P.; Tai, C. Mesh segmentation with concavity-aware fields. IEEE Trans. Vis. Comput. Graph.
**2011**, 18, 1125–1134. [Google Scholar] - Tong, W.; Yang, X.; Pan, M.; Chen, F. Spectral mesh segmentation via L
_{0}gradient minimization. IEEE Trans. Vis. Comput. Graph.**2020**, 26, 1807–1820. [Google Scholar] - Ji, Z.; Liu, L.; Chen, Z.; Wang, G. Easy mesh cutting. Comput. Graph. Forum
**2006**, 25, 283–291. [Google Scholar] [CrossRef] - Zheng, Y.; Tai, C.; Au, O.K. Dot scissor: A single-click interface for mesh segmentation. IEEE Trans. Vis. Comput. Graph.
**2011**, 18, 1304–1312. [Google Scholar] [CrossRef] [Green Version] - Kalogerakis, E.; Hertzmann, A.; Singh, K. Learning 3D mesh segmentation and labeling. ACM Trans. Graph.
**2010**, 29, 102. [Google Scholar] [CrossRef] - Benhabiles, H.; Lavoué, G.; Vandeborre, J.; Daoudi, M. Learning boundary edges for 3D-mesh segmentation. Comput. Graph. Forum
**2011**, 30, 2170–2182. [Google Scholar] [CrossRef] [Green Version] - Wang, Y.; Gong, M.; Wang, T.; Cohen-Or, D.; Zhang, H.; Chen, B. Projective analysis for 3D shape segmentation. ACM Trans. Graph.
**2013**, 32, 192. [Google Scholar] [CrossRef] - Guo, K.; Zou, D.; Chen, X. 3D mesh labeling via deep convolutional neural networks. ACM Trans. Graph.
**2015**, 35, 3. [Google Scholar] [CrossRef] - Kalogerakis, E.; Averkiou, M.; Maji, S.; Chaudhuri, S. 3D shape segmentation with projective convolutional networks. In Proceedings of the 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Honolulu, HI, USA, 21–26 July 2017; pp. 6630–6639. [Google Scholar]
- Xu, H.; Dong, M.; Zhong, Z. Directionally convolutional networks for 3D shape segmentation. In Proceedings of the 2017 IEEE International Conference on Computer Vision (ICCV), Venice, Italy, 22–29 October 2017; pp. 2717–2726. [Google Scholar]
- George, D.; Xie, X.; Tam, G.K. 3D mesh segmentation via multi-branch 1D convolutional neural networks. Graph. Model.
**2018**, 96, 1–10. [Google Scholar] [CrossRef] [Green Version] - Meyer, M.; Desbrun, M.; Schroder, P.; Barr, A.H. Discrete differential-geometry operators for triangulated 2-manifolds. In Proceedings of the 3rd International Workshop Visualization and Mathematics (VisMath), Berlin, Germany, 22–25 May 2002; pp. 35–57. [Google Scholar]
- Shapira, L.; Shamir, A.; Cohen-Or, D. Consistent mesh partitioning and skeletonisation using the shape diameter function. Vis. Comput.
**2008**, 24, 249–259. [Google Scholar] [CrossRef] - Hilaga, M.; Shinagawa, Y.; Komura, T.; Kunii, T.L. Topology matching for fully automatic similarity estimation of 3D shapes. In Proceedings of the 28th Annual Conference on Computer Graphics and Interactive Techniques (SIGGRAPH), New York, NY, USA, 12–17 August 2001; pp. 203–212. [Google Scholar]
- Ben-Chen, M.; Gotsman, C. Characterizing shape using conformal factors. In Proceedings of the 1st Eurographics Workshop on 3D Object Retrieval (3DOR@Eurographis), Hersonissos, Greece, 15 April 2008; pp. 1–8. [Google Scholar]
- Sun, J.; Ovsjanikov, M.; Guibas, L.J. A concise and provably informative multi-scale signature based on heat diffusion. Comput. Graph. Forum
**2009**, 28, 1383–1392. [Google Scholar] [CrossRef] - Chazelle, B.; Dobkin, D.P.; Shouraboura, N.; Tal, A. Strategies for polyhedral surface decomposition: An experimental study. Comput. Geom.
**1997**, 7, 327–342. [Google Scholar] [CrossRef] [Green Version] - Zhou, Y.; Huang, Z. Decomposing polygon meshes by means of critical points. In Proceedings of the 10th International Multimedia Modeling Conference (MMM), Brisbane, QLD, Australia, 5–7 January 2004; pp. 187–195. [Google Scholar]
- Lavoué, G.; Wolf, C. Markov random fields for improving 3D mesh analysis and segmentation. In Proceedings of the 1st Eurographics Workshop on 3D Object Retrieval (3DOR@Eurographis), Crete, Greece, 15 April 2008; pp. 25–32. [Google Scholar]
- Golovinskiy, A.; Funkhouser, T.A. Randomized cuts for 3D mesh analysis. ACM Trans. Graph.
**2008**, 27, 145. [Google Scholar] [CrossRef] - Chen, X.; Golovinskiy, A.; Funkhouser, T. A benchmark for 3D mesh segmentation. ACM Trans. Graph.
**2009**, 28, 13. [Google Scholar] [CrossRef] - Theologou, P.; Pratikakis, I.; Theoharis, T. Unsupervised spectral mesh segmentation driven by heterogeneous graphs. IEEE Trans. Pattern Anal. Mach. Intell.
**2017**, 39, 397–410. [Google Scholar] [CrossRef] - Zhang, H.; Wu, C.; Deng, J.; Liu, Z.; Yang, Y. A new two-stage mesh surface segmentation method. Vis. Comput.
**2018**, 34, 1597–1615. [Google Scholar] [CrossRef] - Zhang, L.; Guo, J.; Xiao, J.; Zhang, X.; Yan, D. Blending surface segmentation and editing for 3D models. IEEE Trans. Vis. Comput. Graph.
**2022**, 28, 2879–2894. [Google Scholar] [CrossRef] - Lin, C.; Liu, L.; Li, C.; Kobbelt, L.; Wang, B.; Xin, S.; Wang, W. SEG-MAT: 3D shape segmentation using medial axis transform. IEEE Trans. Vis. Comput. Graph.
**2022**, 28, 2430–2444. [Google Scholar] [CrossRef] - Zheng, Y.; Tai, C. Mesh decomposition with cross-boundary brushes. Comput. Graph. Forum
**2010**, 29, 527–535. [Google Scholar] [CrossRef] [Green Version] - Fan, L.; Liu, L.; Liu, K. Paint mesh cutting. Comput. Graph. Forum
**2011**, 30, 603–612. [Google Scholar] [CrossRef] - Hou, Y.; Zhao, Y.; Shan, X. 3D mesh segmentation via L
_{0}-constrained random walks. Multim. Tools Appl.**2021**, 80, 24885–24899. [Google Scholar] [CrossRef] - Shu, Z.; Qi, C.; Xin, S.; Hu, C.; Wang, L.; Zhang, Y.; Liu, L. Unsupervised 3D shape segmentation and co-segmentation via deep learning. Comput. Aided Geom. Des.
**2016**, 43, 39–52. [Google Scholar] [CrossRef] - Yi, L.; Su, H.; Guo, X.; Guibas, L.J. SyncSpecCNN: Synchronized spectral CNN for 3D shape segmentation. In Proceedings of the 2017 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Honolulu, HI, USA, 21–26 July 2017; pp. 6584–6592. [Google Scholar]
- Wang, Z.; Lu, F. VoxSegNet: Volumetric CNNs for semantic part segmentation of 3D shapes. IEEE Trans. Vis. Comput. Graph.
**2020**, 26, 2919–2930. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hu, Z.; Bai, X.; Shang, J.; Zhang, R.; Dong, J.; Wang, X.; Sun, G.; Fu, H.; Tai, C. Voxel-mesh network for geodesic-aware 3D semantic segmentation of indoor scenes. IEEE Trans. Pattern Anal. Mach. Intell.
**2022**. [Google Scholar] [CrossRef] - Shu, Z.; Shen, X.; Xin, S.; Chang, Q.; Feng, J.; Kavan, L.; Liu, L. Scribble-based 3D shape segmentation via weakly-supervised learning. IEEE Trans. Vis. Comput. Graph.
**2020**, 26, 2671–2682. [Google Scholar] [CrossRef] [Green Version] - Shu, Z.; Yang, S.; Wu, H.; Xin, S.; Pang, C.; Kavan, L.; Liu, L. 3D shape segmentation using soft density peak clustering and semi-Supervised learning. Comput. Aided Des.
**2022**, 145, 103181. [Google Scholar] [CrossRef] - Xu, X.; Liu, C.; Zheng, Y. 3D tooth segmentation and labeling using deep convolutional neural networks. IEEE Trans. Vis. Comput. Graph.
**2019**, 25, 2336–2348. [Google Scholar] [CrossRef] - Lawonn, K.; Meuschke, M.; Wickenhöfer, R.; Preim, B.; Hildebrandt, K. A geometric optimization approach for the detection and segmentation of multiple aneurysms. Comput. Graph. Forum.
**2019**, 38, 413–425. [Google Scholar] [CrossRef] - Wang, W.; Yu, R.; Huang, Q.; Neumann, U. SGPN: Similarity group proposal network for 3D point cloud instance segmentation. In Proceedings of the 2018 IEEE Conference on Computer Vision and Pattern Recognition (CVPR), Salt Lake City, UT, USA, 18–22 June 2018; pp. 2569–2578. [Google Scholar]
- Hoang, L.; Lee, S.-H.; Kwon, K.-R. A deep learning method for 3D object classification and retrieval using the global point signature plus and deep wide residual network. Sensors
**2021**, 21, 2644. [Google Scholar] [CrossRef] - Liu, B.; Wang, W.; Zhou, J.; Li, B.; Liu, X. Detail-preserving shape unfolding. Sensors
**2021**, 21, 1187. [Google Scholar] [CrossRef] [PubMed] - Liu, Z.; Xiao, X.; Zhong, S.; Wang, W.; Li, Y.; Zhang, L.; Xie, Z. A feature-preserving framework for point cloud denoising. Comput. Aided Des.
**2020**, 127, 102857. [Google Scholar] [CrossRef] - Liu, Z.; Li, Y.; Wang, W.; Liu, L.; Chen, R. Mesh total generalized variation for denoising. IEEE Trans. Vis. Comput. Graph.
**2022**, 28, 4418–4433. [Google Scholar] [CrossRef] [PubMed] - Hou, Y.; Zhao, Y. A robust segmentation algorithm for 3D complex meshes. In Proceedings of the 7th International Conference on Computer-Aided Design, Manufacturing, Modeling and Simulation (CDMMS), Busan, Republic of Korea, 14–15 November 2020; p. 032045. [Google Scholar]
- Jones, R.T.; Durand, F.; Desbrun, M. Non-iterative, feature-preserving mesh smoothing. ACM Trans. Graph.
**2003**, 22, 943–949. [Google Scholar] [CrossRef] [Green Version] - Zheng, Y.; Fu, H.; Kin-Chung Au, O.; Tai, C. Bilateral normal filtering for mesh denoising. IEEE Trans. Vis. Comput. Graph.
**2011**, 17, 1521–1530. [Google Scholar] [CrossRef] - Pauly, M.; Gross, M.H.; Kobbelt, L. Efficient simplification of point-sampled surfaces. In Proceedings of the 13th IEEE Visualization Conference (IEEE Vis), Boston, MA, USA, 27 October–1 November 2002; pp. 163–170. [Google Scholar]
- Boykov, Y.; Kolmogorov, V. An experimental comparison of Min-Cut/Max-Flow algorithms for energy minimization in vision. IEEE Trans. Pattern Anal. Mach. Intell.
**2004**, 26, 1124–1137. [Google Scholar] [CrossRef] [Green Version] - Katz, S.; Leifman, G.; Tal, A. Mesh segmentation using feature point and core extraction. Vis. Comput.
**2005**, 21, 649–658. [Google Scholar] [CrossRef] - Lai, Y.; Hu, S.; Martin, R.R.; Rosin, P.L. Fast mesh segmentation using random walks. In Proceedings of the 2008 ACM Symposium on Solid and Physical Modeling, New York, NY, USA, 2–4 June 2008; pp. 183–191. [Google Scholar]
- Attene, M.; Falcidieno, B.; Spagnuolo, M. Hierarchical mesh segmentation based on fitting primitives. Vis. Comput.
**2006**, 22, 181–193. [Google Scholar] [CrossRef] [Green Version] - Chen, L.; Georganas, N.D. An efficient and robust algorithm for 3D mesh segmentation. Multim. Tools Appl.
**2006**, 29, 109–125. [Google Scholar] [CrossRef]

**Figure 1.**Algorithm overview conducted on an armadillo mesh [6]. Firstly, we perform a rapid over-segmentation using the adaptive space partition to yield a set of superfacets. Secondly, we use a feature-aware region fusion method to merge similar superfacets to generate the final segmentation result. For demonstration, all superfacets and segmentation parts are rendered with random colors.

**Figure 2.**Over-segmentation carried out on a horse mesh. (

**a**) The original mesh. (

**b**) Over-segmentation result.

**Figure 3.**Segmentation boundary smoothing. Please note that jaggy boundaries become smoother. (

**a**) Boundaries before smoothing. (

**b**) Boundaries after smoothing.

**Figure 6.**Segmentation results on noisy goblet meshes. The meshes in (

**a**–

**c**) are corrupted by Gaussian noises along normal directions with intensities of 0.3, 0.4, and 0.5, respectively. The meshes in (

**d**–

**f**) are corrupted by Gaussian noises along random directions with intensities of 0.3, 0.4, and 0.5, respectively.

**Figure 10.**Segmentation results of airplane meshes with different samplings. (

**a**) A low-sampling mesh rendered in wireframe (5400 vertices and 10,796 facets). (

**b**) A middle-sampling mesh rendered in wireframe (9417 vertices and 18,830 facets). (

**c**) A high-sampling mesh rendered in wireframe (19,533 vertices and 39,062 facets). (

**d**) Result of (

**a**). (

**e**) Result of (

**b**). (

**f**) Result of (

**c**).

**Figure 13.**Quantitative comparisons with five state-of-the-art segmentation algorithms of [14,48,49,50,51]. RI is Rand Index, CD is Cut Discrepancy, GCE is Global Consistency Error, LCE is Local Consistency Error, Hamming is Hamming Distance, Hamming-Rm is missing rate, and Hamming-Rf is false alarm rate. Please note that lower values indicate better results. (

**a**) Rand Index. (

**b**) Cut Discrepancy. (

**c**) Consistency Error. (

**d**) Hamming Distance.

**Figure 14.**Segmentation results of a teddy mesh with different parameters. Apparent errors appear in (

**b**). Please note that the parameters should be set properly to produce a satisfactory result. (

**a**) Our result with {${q}_{max}$ = 40, ${\sigma}_{max}$ = 0.1, $\mathrm{m}$ = 8}. (

**b**) Our result with {${q}_{max}$ = 5, ${\sigma}_{max}$ = 0.05, $\mathrm{m}$ = 1.9}.

Mesh | Number of Vertices | Number of Facets | Number of Superfacets | Time |
---|---|---|---|---|

Airplane | 5400 | 10,796 | 389 | 0.0844 |

Armadillo | 25,273 | 50,542 | 1311 | 1.1754 |

Bird | 7849 | 15,694 | 587 | 0.1507 |

Cup | 15,198 | 30,396 | 1307 | 0.2736 |

Hand | 7112 | 14,220 | 536 | 0.1298 |

Horse | 48,485 | 96,966 | 1218 | 1.5426 |

Octopus | 5944 | 11,888 | 412 | 0.1013 |

Teddy | 11,090 | 22,176 | 329 | 0.0964 |

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**MDPI and ACS Style**

Wu, L.; Hou, Y.; Xu, J.; Zhao, Y.
Robust Mesh Segmentation Using Feature-Aware Region Fusion. *Sensors* **2023**, *23*, 416.
https://doi.org/10.3390/s23010416

**AMA Style**

Wu L, Hou Y, Xu J, Zhao Y.
Robust Mesh Segmentation Using Feature-Aware Region Fusion. *Sensors*. 2023; 23(1):416.
https://doi.org/10.3390/s23010416

**Chicago/Turabian Style**

Wu, Lulu, Yu Hou, Junli Xu, and Yong Zhao.
2023. "Robust Mesh Segmentation Using Feature-Aware Region Fusion" *Sensors* 23, no. 1: 416.
https://doi.org/10.3390/s23010416